Distribution of values of the sum of unitary divisors in residue classes Текст научной статьи по специальности «Математика»

Probl. Anal. Issues Anal. Vol. 5(23), No. 1, 2016, pp. 31-44

DOI: 10.15393/j3.art.2016.3370

31

UDC 511

B. M. Shirokov, L. A. Gromakovskaya

DISTRIBUTION OF VALUES OF THE SUM OF UNITARY DIVISORS IN RESIDUE CLASSES

Abstract. In this paper we prove the tauberian type theorem containing the asymptotic series for the Dirichlet series. We use this result to study distribution of sum of unitary divisors in residue classes coprime with a module. The divisor d of the integer n is an unitary divisor if ^d, ^ =1. The sum of unitary

divisors of a number n is denoted by a*(n). For a fixed function f (n) let us denote by S(x, r) the numbers of positive integers n < x such that f (n) = r (mod N) for x > 0 and r coprime with module N. According to W. Narkiewicz [5], a function f (n) is called weakly uniformly distributed modulo N if and only if for every pair of coprime integer a, b

l« S(x, a) i x^x S(x, b)

provided that the set {r | (r, N) = 1} is infinite. We use the tauberian theorem to obtain an asymptotic series for S(x, r) for a*(n). Then we derive necessary and sufficient conditions for the module N that provide weakly uniform distribution modulo N of the function a* (n).

Key words: sum of the unitary divisors, tauberian theorem, distribution of values in the residue classes

2010 Mathematical Subject Classification: 11N69

1. Introduction. Problems connected with distribution of values of arithmetical functions in residue classes have been considered from the middle of the XX century. Originally the purpose of these papers was to obtain asymptotics of the value set connected by the some congruent condition, mainly on a prime module (see, for example, [1, 2]).

©Petrozavodsk State University, 2016

Paper by I. Niven [3] contains the concept of uniform distribution in residue classes and in ring Z and started studying conditions of uniform distribution in residue classes. S. Uchiyama [4] obtained necessary and sufficient conditions for uniform distribution of sequence on given module and established duality of this concept to the concept of uniform distribution of a sequence modulo 1.

In paper [5] W. Narkiewicz transfered this concept to the residue classes coprime with a module. He called uniformly distribution in these classes weakly uniform and deduced test for weakly uniform distribution for some class of multiplicative functions. Using this test J. Sliva [6] obtained necessary and sufficient conditions on module that sum of divisors is weekly uniformly distributed on this module.

Let N be some fixed integer. For some fixed function f (x) let us denote by S(x, r) the natural number n < x for which f (n) = r (mod N) for x > 0 and for r coprime with some module N.

According to W. Narkiewicz [5] a function f (n) is called weakly uniformly distributed modulo N if and only if for every pair of coprime integer a, b

... s (x, a)

lim g/ n = 1

x^ro S (x, b)

provided that the set {r | (r,N) = 1} is infinite.

The results of W. Narkiewicz [5] and J. Sliva [6] are transfered to a more wide class of multiplicative functions in [7]-[9].

Let us note the paper of O. M. Fomenko [10] that contains results on distribution of values in residue classes for functions which do not belong to Narkiewicz classes.

The asymptotic formulae for S(x, r) in the papers contain only the main member of asymptotics of weekly uniform distribution of multiplicative functions. These results can be sharpen in most cases; this was done in [7] - [9]. The point is that usually the Delange tauberian theorem [11] is applied to generating the Dirichlet series. But generating functions used by this theorem have better properties than the theorem demands.

We prove the tauberian type theorem 1 in this paper, which can replace the Delange theorem to obtain sharper results.

This paper is devoted to study the weekly uniform distribution of values of sum of unitary divisors denoted by a* (n).

Definition 1. A divisor d of n is called unitary divisor if ^n, ^ = 1.

Upper bound set of values of the function a*(n) was studied in [12].

Let us introduce necessary notation. Let N, a, b, d, i, j, k, l, m, n and r be nonnegative integers; p, q be prime numbers; x, X be the Dirichlet characters and xo, X0 be the relevant principal characters; (m, n) be the greatest common divisor of integers m and n; G(N) be the residue multiplicative group modulo N. We use notation S(x, r) for the function a*(n) in the sequel.

On the complex s-plane, s = a + it, for the some constants c0 > 0 we denote a(t) = 1 — -—, ,, , — oo < t < oo and

() ln(2 + |t|r

= {s = a + it | a > max | a(t), 31 , — to < t < +to}; w(n) is the number of distinct prime divisors of n;

q — 2

* = n q-2 ■ ^ = II

1 5 h" — V -V II 1 )

q 1 q 1

q\N *

if N = q^1 ■ ■ ■ qmm then let us denote N = qi ■ ■ ■ qm; let ^(n) be Euler's function.

In this paper we prove the following theorems.

Theorem 1. Let for a complex-valued function f (n) a number A > 0 exist, such that f (n) = O(lnA n) and function F(s) defined for a > 1 as sum of the series

" f (n)

F(s) = £

n=1

satisfy the following conditions:

1) there are a complex number z and analytical in Q function G(s) such that G(1) = 0 and F (s) can be extended to the region Q by the equality

F (s) = (S^Sy, s G Q(a). (1)

2) there is a constant c1 > 0 such that for the extension F (s) the estimate

F (s)= O(lnC1 (2 + |t|), |t|> 1, (2)

is correct.

Then there is a number c > 0 such that

a) if z = 0, -1, -2,... then

£ f (n) = O(xe-c^); (3)

b) if z = 1 then

£ f (n) = G(1)x + O(xe-c7inX); (4)

c) if z is not integer then for any positive integer n

£ f (n) = (ln X)1-z ^^ mi) + O{ (ln x)nX+1-Re z) , (5)

where

" (z - 1) ■ ■ ■ (z - k) y k

r(z)

pn(y) = ^ a^--- y

k=1

r(z) is the Euler gamma-function, ak are Taylor coefficients of the

■ G(s) .

function - in point s =1.

s

Theorem 2. If N is an odd number then for any positive integer n and for any character x modulo N there are polynomials Pn(y) and Qn(y, x) such that for any r E G(N) for x ^ to

S(x, r) = X

p(N)

1 fpj +

(ln x)1-A \ \ ln XJ \(ln x)n+1

+ £ nnxW\Qn{ lnrx»^ +O 1

(ln x)1 ^ \ \ ln x' / \ (ln x)n+1

X=X0

X^(0N} =X0

Summation is taken over the nonprincipal characters which have order p(N) in the character group modulo N in last sum.

Theorem 3. The function a*(n) is weakly uniformly distributed modulo N if and only if N is odd.

2. Proof of Theorem 1. We apply the approximate Perron formula with the half-integer x for the sum of f (n) values (see [13, p. 427]). Let

2 _

o0 = 1 + ,- and T = ev'1nX. Let us choose any c E (0,1). Then

In x

lnA x • e—v'inx = ). The Perron formula implies

ao+iT

^f (n) = _L ^ F(s)XS ds + O(xe—cVTnx). (6)

n<X TTI

— ao —iT

Let us apply the lemma from paper [7, p. 180] (see also [14, p. 14]) to the right-hand side integral. This lemma states that provided that F(s) obeys assumptions of the theorem asymptotic formulae listed in the statement of the theorem hold for the integral. This proves Theorem 1.

3. Proof of Theorem 2. Let N = q^1 • • • q^ and qi > 2. Using properties of characters we can present S(x, r), r E G(N), in the following form:

S(x, r) = £ 1 = -Ny £ 0x(r) £ x(o*(n)).

n—x X n—x

a* (n) =r (mod N)

Let us denote

S(x,X) = £ X(o*(n)).

n— x

Then S(x, r) takes the form

11

S(x r) = S(x xo) + £ x(r)S(x X). (7) -(N) -(N) xTXo

The sum S(x, x) is sum of coefficients of the Dirichlet series:

F(s.x) = £ ^ (8)

n=l

Thus the problem is reduced to applying Theorem 1 to the Dirichlet series (8). To do this we need to study analytic properties of the function F(s, x) in the domain fi.

The function F(s, x) is analytic in the half-plane o > 1 as |x(n)| < 1. Since the series (8) converges absolutely for o > 1 but the function o*(n) is multiplicative and o*(pk) = pk + 1, the Euler theorem implies

F (s,x) = n(1 + ^ + ^ +

Let us denote

A1 (s, x) = (1 -

xp)(>+£

k=1

+ > x(2k + 1^ ^ L + £ x(pk + 1)

2ks

A2 (s, x) = n 1 + £

p\N \ k = 1

x(pk + 1) - x(p + 1)x(pk-1 + 1)

pk

p\ N p> 2

k=2

P

ks

H (s,x)^(1

p\N

x(P + 1)

Ps

1

and assume that A(s, x) = A1 (s, x) ■ A2(s, x). Then for a > 1 the function F(s, x) takes the form

F(s, x) = H(s,x) ■ A(s,x).

This representation allows extension of F(s, x) to the domain fi. Consider the function A(s, x). The factor

1

xp) n (!+£

7 p\N \ k=1

x(pk + 1)

P

ks

is the function that is regular, bounded, and does not vanish for a > 3/4. Product taken over p | N does not vanish in the same half-plane. The sum of the series

x(2k + 1)

1+

£

k=1

2ks

can vanish for s = 1 if and only if x(2k + 1) = -1 for any natural k. But it is impossible because if x(3) = -1 then x(23 + 1) = x(9) = x2(3) = 1. For sum of the series in the product A2(s, x) for a > 3/4 the estimation

£

k=2

x(pk + 1) - x(p + 1)x(pk 1 + 1)

P

ks

<

2

<

2

pa (p° - (3ff - 1)'

is correct. The number in the right-hand side is less that 1 since 33/4 > > 2. Thus each member of the product A2 (s, x) does not vanish and the product converges absolutely for a > 3/4. Therefore the function A2(s, x) is regular, bounded in region fi and does not vanish for s = 1.

Let us consider ln H(s, x) and note that the series

x(p + 1)

5> 1

Ps

p\N

converges absolutely for a > 1 because the series composed from the real and imaginary parts of this series converge absolutely; all logarithm branches are important here. Therefore for some branch we have

Ln nil - *<P±i) V1 = - £ ln A - x(p +1>

ps J , V ps

p\ N p\ N

Therefore

Ln H (s, x) = £x(p^ + ££x^ (9)

p 1 p k>2 1

Let us denote by B(s, x) the sum of the series for k > 2. This function is regular and bounded for a > 3/4. In fact, the estimate

£ Xk (p + 1)

k< 2 pkS

< £ -7—1-TV = O(p-3/2)

- Z_2 p<r (p* - 1) ^ ;

implies that the series converges absolutely and uniformly in the described domain.

The first sum taken over prime numbers in equality (9) contains necessary information. Let us transform it using the character properties:

y- X(P + 1) = y- y- X(P + 1) = y- y-S(pr) X(P + 1)

/ J pS / J / J pS / J / J vF> ) pS '

p 1 reO(N) p=r (mod N) reO(N) p 1

where 5(p, r) is equal to 1 if p (mod N) = r and equal to 0 otherwise. The orthogonality property of characters X(n) is the following formula:

s(p >r)=^N) £ x^(r)x (p).

X

Therefore

£ ^ = £ ¿No £ x(r+D-X(r) £ Xf- (10)

p 1 X rv > reC(N) P 1

Let us denote

z(x,X ) =

-(N)

£ x(r + 1)X(r).

(11)

reG(N)

The sum in the right-hand side of (10) is over prime numbers and contains

all necessary information about the Dirichlet L-function L(S,X): X(p) , r, V-X(pk)

T,

ps

lnL(s,X) -££

p k>2

p

ks

(12).

The logarithm branch is selected in the same way as on the page 37. Let

X (pk)

C (s,X ) = -EE

p k>2

ps

(13)

This function is also regular and bounded for o > 3/4: this is shown in the

same way as for B(s, x). Equalities (10)-(13) allow representing F(s,x) in the following form:

F(s, x) = A(s,x)eB(s'x) H ez(x'X)C(s'X) • [l(s,X)

<x,X )

(14)

X

It is well known that functions L(s, X) for every X = X0 and (s — 1)L(s,X0) are regular in fi and do not vanish for s = 1 (see for example [13, ch. IV]). Let us denote

z (x) = z (x,Xo).

(15).

Thus the function

G(s, x) A(s, x)eB(s'X^ ez(x'X)C(s'X) ^ L(s, X)

X=Xo

X

Kx,X)

X

X

(s — 1)L(s,Xo)

z(x)

is regular in fi and G(1,x) = 0. Thus equalities (14) and (16) imply

F(s, x) =

G(s,x)

(s — 1)z(X) ■

1

Last equality allows extension of the function F (s, x) to fi so that F (s, x) satisfies condition (T) of Theorem 1 with exponent z = z(x).

It is known that L(s, x) = O(ln(2 + |t|) in fi for t ^ to (see [13,

p. 132]). Equality (14) and properties of functions A(s, x), B(s, x) and

C(s, x) imply that there is a constant c1 > 0 such that

F(s, x) = O(lnC1 (2 + |t|)) (17)

in fi for |t| ^ TO.

This estimate shows that function F(s, x) satisfies the condition (2). The last step to make is to calculate the exponent z(x) for each character modulo N.

Let Ni = qk. Then N = N1 • • • Nm and G(N) = G(N) x • • • x G(Nm). The character x(n) modulo N is defined by equality

x(n) = xi(n) ••• x™(n), (18)

where xi is the character modulo Ni. Since the Euler function is multi-

plicative equality (15) implies

m 1 m

z(x) = n ^rm £ xi(r+1) = n z(xi). a^

i=i -( i) reO(Ni) i=i

Let us calculate the exponent z(x) for the character x(n) modulo N = = qk . There are two possibilities, one of them is divided in two subcases.

1. x = xo.

Every r E G(N) is represented uniquely in the form

r = aq + 6, 0 < a < qk—1 — 1, 1 < b < q — 1. (20)

Consider the transformation r ^ r + 1. Each residue aq + q — 1 turns to residue aq + q on which the character vanishes. Besides, residues aq +1 for each a disappear after such transformation. Thus the number of residues r + 1 for which x0 = 1 equals qk—1 (q — 2). Therefore

qk—1 (q — 2) q — 2

z(xo) = -(n) = q—1. (21)

2. x = xo.

Using characters properties, in the same way as above, we conclude that

qk — 1 — 1 qk-1 -1

Y x(r + 1) = Y x(r) - Y x(aq + 1) = - Y x(aq + 1).

reC(N) reC(N) a=0 a=0

Let us calculate the last sum. Let g be the primitive root modulo qk and l = indg r for r = aq +1. This means that r = gl = 1 (mod q). Then l = 0 (mod(q - 1)). Therefore an integer n, 0 < n < qq—1 - 1 corresponds to each coefficient a one-to-one so that

aq +1 = gn(q—1) (mod qk). (22)

The cyclic group character x is defined uniquely by the value x(g) as one

of the values of ^T. Therefore the equality (22) implies

x(aq +1) = (x(g))n(q—1}. For the brevity of notation let us assume Z = (x(g))n(q—1). Then

. qk-1 —1

1 x ^n

z(x) = - *N) „10 C'. (23)

a) Z = 1.

This means that the character x has order q - 1 in the character group modulo N. Then

qk—1 1

z(x) = - ~7~~k\ =--r. (24)

p(qk) q - 1

b) Z = 1.

In this case

1 - zqfe-1

z(x) = - WW-O =0 , (25)

since Zq'-1 = (x(g))"(N) = 1.

Now on the base of the formula (19) for N = N1 ■ ■ ■ Nm we obtain

three possible forms of the exponent z(x):

1) x = xo- In this case

z(xo) = n q-! = a.

q\NH

2) x = Xo, but x^(N) = Xo. Then

z(x) = ( -i)-(N) n A =*

q\N q

3) There is a factor Xi in the produc x = Xi''' Xm that Xq-1 = (Xi)0, i.e., x^(N) = X0- Then z(x) = 0.

In the first case z(x0) = A is not integer. Then Theorem 1 implies that for every positive integer n there is a polynomial Pn (y) of power n such that

S(x-xo)=(l^ (P" (l^x) + O ((hT^)). (26)

In the second case, i.e., a character x = X0 exists and has order ^(NV) but z(x) = * is not integer, Theorem 1 also implies existence of a polynomial Q(y, x) of power n for all positive integer n such that

S(Q (¿*) + O ((dpi)). (27)

In the third case, when z(x) = 0, Theorem 1 implies

S(x, x) = O(xe-c^inX). (28).

Now substituting S(x, x) for all forms of z(x) to formula (7), we obtain asymtotic equality of Theorem 2. This finishes the proof of Theorem 2.

4. Proof of Theorem 3. The proof is based on the following test by W. Narkiewicz [5].

A function f (n) is weekly uniformly distributed (mod N) if and only if for every nonprincipal character x (mod N) the following evaluation holds

£ x(f (n))= o (£ X0(f (n)) (29)

n<x n<x

for x .

If N is odd then formulae (26)-(28) imply equality (30). Therefore if N is odd then a* (n) is weekly uniformly distributed. Let N be even. If pa | N, p > 2, a > 1 then

Ex(f (n))= o(£xo(f (n)

n<x n<x

but pa + 1 is even. Therefore

x(a* (n))=0 for every character modulo N. Therefore

S(x,x) = £ x(a*(n))= £ x(2k + 1).

(30)

nx

2k < a

Let N = 2m, m > 2. For k > m and for every character x (mod N) we have x(2k + 1) = 1, since the congruence 2k + 1 = 1 (mod N) holds. Therefore for every character x (mod N)

m — 1

S(x,x)=£ x(2k + 1)+ ^ 1

k=1

,, , In x m<k< 1^2

ln x

+ O(1)

Thus equality (30) is wrong.

Let now N = 2m Q, m > 1 and Q be odd. If for every x = x0 equality (30) holds then

£ S(x,x) = o(S(x,xo^.

X=X0

Therefore

£S(x,x) ~ S(x,xo).

But it is wrong. In fact,

£ S(x,x)= £ £x(2k + 1).

In x X

(31)

k< In 2

If (2k + 1,Q) > 1 then the last sum vanishes. Therefore the sum for ln x

k < -—- in equality (32) contains only members for which (2k +1, N) = 1.

But if (n, N) = 1 then

<p(N), n = 1 (mod N),

£x(n) =

0, n = 1 (modN).

But the congruence 2k + 1 = 1 (mod N) implies the congruence 2k + 1 = = 1 (mod Q), that is 2k = 0 (mod Q); this is impossible for any natural k. Therefore

£S (x,x) = 0. x

Formula (31) implies equality S(x,x0) = 0 since the values of this sum are integer. This equality implies x0(2k + 1) = 0 for every natural k since the values of the principal character are 0 and 1. But it means that for every k

dk = (2k + 1, Q) > 1. (33)

But for every integer l

2MQ) + 1 = 2 (mod Q).

for k = l^(Q). This congruence holds for module dk since the number Q is divided by dk for every k. In other words, dk is coprime with the

number 2k + 1 but this contradicts to inequality (33). This contradiction

proves that relation (31) is wrong. Therefore the function is not weekly

uniformly distributed. Theorem 3 is proved.

References

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Received May 11, 2016. In revised form, October 8, 2016. Accepted October 8, 2016.

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